Electronic edition published by Cultural Heritage Langauge Technologies (with permission from Charles Scribners and Sons) and funded by the National Science Foundation International Digital Libraries Program. This text has been proofread to a low degree of accuracy. It was converted to electronic form using data entry.

LEIBNIZ, GOTTFRIED WILHELM (b. Leipzig,
Germany, 1 July 1646; d. Hannover, Germany,
14 November 1716), mathematics, philosophy, metaphysics.

of partial differentiation; the reduction of the differential
equation

a00 + a10x + (a01 +
a11x)y' = 0

by means of the transformation

x = p11u + p12v, y = p21u +
P22v;

the solution of

y' + p(x)y + q(x) = 0

and

(y')2 + p(x)y' + q(x) = 0

through series with coefficients in number couples;
and the reduction of the equation, which had originated
with Jakob Bernoulli,

y' = p(x)y + q(x)yn

to

y' = P(x)y + Q(x).

Leibniz' inventive powers and productivity in
mathematics did not begin to slow until around 1700.
The integration of rational functions (1702-1703) was,
to be sure, an important accomplishment; but the
subject was not completely explored, since Leibniz
supposed (Johann Bernoulli to the contrary) that there
existed other imaginary units besides ?-1 (for
example, 4?-1) which could not be represented by
ordinary complex numbers. The subsequent study,
which was not published at the time, on the integration
of special classes of irrational functions also remains
of great interest. The discussion with Johann Bernoulli
on the determination of arclike algebraic curves in the
plane (1704-1706) resulted in both a consideration of
relative motions in the plane and an interesting
geometric construction of the arclike curve equivalent
to a given curve. This construction is related to the
optical essay of 1689 and to the theory of envelopes
of 1692-1694, but it cannot readily be grasped in
terms of a formula (1706). The remarks (1712) on the
logarithms of negative numbers and on the representation
of ?(-1)n by 1/2 can no longer be
considered satisfactory. The description of the
calculating machine (1710) indicates its importance
but does not give the important details; the first
machines of practical application were constructed by
P. M. Hahn in 1774 on the basis of Leibniz' ideas.

Leibniz accorded a great importance to mathematics
because of its broad interest and numerous
applications. The extent of his concern with mathematics
is evident from the countless remarks and
notes in his posthumous papers, only small portions of
which have been accessible in print until now, as well
as from the exceedingly challenging and suggestive
influential comments expressed brilliantly in the
letters and in the works published in his own lifetime.

Leibniz' power lay primarily in his great ability
to distinguish the essential elements in the results of
others, which were often rambling and presented in a
manner that was difficult to understand. He put them
in a new form, and by setting them in a larger context
made them into a harmoniously balanced and
comprehensive whole. This was possible only because
in his reading Leibniz was prepared, despite his
impatience, to immerse himself enthusiastically and
selflessly in the thought of others. He was concerned
with formulating authoritative ideas clearly and connecting
them, as he did in so exemplary a fashion in
the mathematics of infinitesimals. Interesting details
were important—they occur often in his notes—but
even more important were inner relationships and
their comprehension, as this term is employed in the
history of thought. He undertook the work required
by such an approach for no other purpose than the
exploration of the conditions under which new ideas
emerge, stimulate each other, and are joined in a
unified thought structure.

JOSEPH E. HOFMANN

BIBLIOGRAPHY

I. ORIGINAL WORKS.

The following volumes of the
Sämtliche Schriften und Briefe, edited by the Deutsche
Akademie der Wissenschaften in Berlin, have been published:
Series I (Allgemeiner politischer und historischer
Briefwechsel), vol. 1: 1668-1676 (1923; Hildesheim, 1970);
vol. 2: 1676-1679 (1927; Hildesheim, 1970); vol. 3: 1680-1683
(1938; Hildesheim, 1970); vol. 4: 1684-1687 (1950);
vol. 5: 1687-1690 (1954; Hildesheim, 1970); vol. 6: 1690-1691
(1957; Hildesheim, 1970); vol. 7: 1691-1692 (1964);
vol. 8: 1692 (1970); Series II (Philosophischer Briefwechsel),
vol. 1: 1663-1685 (1926); Series IV (Politische Schriften),
vol. 1: 1667-1676 (1931); vol. 2: 1677-1687 (1963);
Series VI (Philosophische Schriften), vol. 1: 1663-1672
(1930); vol. 2: 1663-1672 (1966); vol. 6: Nouveaux essais
(1962).

Until publication of the Sämtliche Schriften und Briefe
is completed, it is necessary to use earlier editions and
recent partial editions. The most important of these
editions are the following (the larger editions are cited
first): L. Dutens, ed., G. W. Leibnitii opera omnia, 6 vols.
(Geneva, 1768); J. E. Erdmann, ed., G. W. Leibniz. Opera
philosophica quae extant latina gallica germanica omnia
(Berlin, 1840; Aalen, 1959); G. H. Pertz, ed., G. W. Leibniz.
Gesammelte Werke, Part I: Geschichte, 4 vols. (Hannover,